# Circumferential Flow in a Cylinder Induced by a Rotating Solid

In this application, AcuSolve is used to simulate the flow of air between concentric cylinders that is initiated by the rotation of the solid inner cylinder. The outer cylinder is held stationary while the inner cylinder rotates with a constant speed. AcuSolve results are compared with analytical results as described in White (1991). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to maintain a continuous velocity across a non-conformal guide surface interface.

## Problem Description

The problem consists of a fluid with material properties of a viscous fluid with the density of air rotating between two infinitely long concentric cylinders, as shown in the following image, which is not drawn to scale. The radius of the outer cylinder is 3.0 m and the radius of the inner cylinder is 1.5 m. The inner cylinder is defined as a solid medium and rotates at 1 rad/sec. The inner cylinder's surface velocity is mapped to the adjacent surface within the fluid medium, while the outer cylinder is fixed. The induced flow is laminar and exhibits a steady state behavior after running for a short period of physical time. The rotation of the inner cylinder causes the fluid to rotate due to viscous shearing near the wall. As the outer cylinder is fixed, a velocity gradient is generated as a function of radius within the flow domain.

The simulation was performed as a two dimensional problem by constructing a volume mesh that contains a single layer of elements in the extruded direction, normal to the flow plane and by imposing slip flow and mesh boundary conditions on the extruded planes.

## AcuSolve Results

The AcuSolve solution reaches a quasi-static result with the velocity between the two cylinders reaching a steady state at the simulation end time. As the solid inner cylinder rotates, it acts as a guide surface for the fluid elements to track and inherit the two velocity components. The velocity at the edge of the inner cylinder is equal to exactly the rotational speed multiplied by the radius. As the radius increases, the flow velocity reduces until it reaches zero at the outer diameter. The following images show the tangential velocity within the fluid as a function of radius from the center. The tangential velocity is compared with the analytical solution. The image below shows black circles representing the analytical solution and a solid red line for the AcuSolve results.

## Summary

The AcuSolve solution compares well with analytical results for laminar flow between concentric cylinders, where the inner cylinder guides the adjacent fluid elements. In this application, a fluid initially at rest inherits the angular velocity from the inner cylinder while the outer cylinder does not rotate. As a result of the inherited velocity and viscosity of the fluid, flow develops between the two cylinders. The AcuSolve solution for the tangential velocity as a function of radius matches exactly with the analytical results.

## Simulation Settings for Flow Between Concentric Split Cylinders

HyperWorks CFD database file: <your working directory>\annulus_split_rotating\annulus_split_rotating.hm

• Problem Description
• Analysis type - Transient
• Flow equation - Navier Stokes
• Turbulence equation - Laminar
• Mesh type - Arbitrary Mesh Movement (ALE)
• Auto Solution Strategy
• Max time steps - 40
• Initial time increment- 0.02 sec
• Convergence tolerance - 0.001
• Min stagger iterations - 2
• Max stagger iterations - 4
• Material Model
• Air
• Density
• Type - Constant
• Density - 1.225 kg/m3
• Viscosity
• Type - Constant
• Viscosity - 1.781 kg/m-sec
• Mesh Motion
• Rotation
• Rotation Center
• X-coordinate - 2.0 m
• Y-coordinate - 0.0 m
• Z-coordinate - 0.0 m
• Angular velocity

Model

• Volumes
• CylinderInner
• Element Set
• Medium - Solid
• Material model - Aluminum
• CylinderOuter
• Element Set
• Medium - Fluid
• Material model - Air
• Surfaces
• InBottom
• Simple Boundary Condition
• Type- Slip
• Mesh displacement type- Slip
• InInterf
• Simple Boundary Condition
• Type- Wall
• Mesh displacement type- Fixed
• Mesh motion- Rotation
• Guide Surface
• Type- Faceted
• InTop
• Simple Boundary Condition
• Type- Slip
• Mesh displacement type- Slip
• OutBottom
• Simple Boundary Condition
• Type- Slip
• Mesh displacement type- Slip
• OutInterf
• Simple Boundary Condition
• Type- Wall
• Wall velocity type- Match Mesh Velocity
• Mesh displacement type- Guide surface
• Guide surface- InInterf
• OutTop
• Simple Boundary Condition
• Type- Slip
• Mesh displacement type- Slip
• OutWall
• Simple Boundary Condition
• Type- Wall
• Mesh displacement type- Fixed
• Nodes
• PressurePoint
• Pressure
• Type- Zero

## References

F. M. White. "Viscous Fluid Flow". Section 3-2.3. McGraw-Hill Book Co., Inc. New York. 1991.